The strong efforts towards the realization of scalable quantum computing are motivated by the promise of super-polynomial speed-up in several computational tasks. A first fundamental step in this route is to provide concrete evidence of ‘quantum supremacy'. Such a regime is obtained when a quantum device is capable of solving a specific problem faster than the best classical hardware. A serious attempt to provide a first demonstration of quantum speed-up with affordable resources is represented by boson sampling [1,2], a non-universal quantum-computational protocol exploiting indistinguishable single photons and linear optical devices. In this context, strong theoretical evidence has been reported that the interference pattern generated by a boson sampler is hard to simulate with classical resources, this pattern being related to the calculation of a hard-to-compute quantity known as a matrix permanent. Formal achievement of the regime of quantum supremacy will answer two very relevant questions: Are we able to benefit from the power of quantum mechanics or does there exist a fundamental reason why this achievement is forbidden? Once this regime has been achieved, are we able to certify the correct functioning of the quantum boson samplers or do we face an undecidability problem? Starting from these intriguing and promising theoretical results, an experimental race kicked off with the aim of reaching the quantum supremacy regime with boson sampling by employing a photonic platform [3]. A large amount of effort has been devoted to pushing the limits of current photonic systems [1], in order to significantly increase the size (number of photons and optical modes) that can be handled with a quantum device. In parallel to this quest for improved quantum performance, it is fundamental to benchmark the limits of a classical simulation of boson sampling. More specifically, in the race for quantum supremacy, what is the best possible classical simulation that can be performed? Such an analysis would be necessary to define the computational capacity possessed by such a classical agent to perform the task (see Fig. 1). In the race for quantum supremacy, a user who employs the best classical hardware and software, and a second agent who exploits a quantum device compete to solve the boson sampling task faster. In a recent article, Wu et al. [4] have provided a fundamental step to define and push the limits of a classical simulation of boson sampling. The authors have employed Tianhe-2, one of the world's most powerful super-computers, to perform such a task. More specifically, by using such high-performance computing hardware and by employing the fastest algorithm to date, Wu et al. have shown that calculating the permanent of a |$50 \times 50$| matrix can be performed in 100 min. Recent results [5] have shown that this computation corresponds to the classical hardness of drawing a 50-photon sample from a boson sampling distribution. The strong computational capacity of Tianhe-2 also allowed us to address the precision that the known fastest classical algorithms can reach in such a large-scale calculation. This latter aspect is indeed of fundamental importance, since any reliable classical computation must be accompanied by an appropriate error analysis. The study reported in Ref. [4] represents by far the largest calculation to date related to the classical simulation of boson sampling, whose relevance can be found in two different but related aspects. First, it raises the bar for any quantum implementation aimed at reaching quantum advantage, and provides a fundamental benchmark to which any photonic hardware must compare. This result suggests that a larger effort must be dedicated to pushing quantum technologies in this direction. Second, improved classical simulation also has important consequences in the context of boson sampling certification, i.e. the verification that a quantum device is correctly performing the required task. Indeed, first validation procedures in the quantum advantage regime must be assisted by powerful classical computational resources that, while requiring longer processing time than quantum hardware, will necessarily have to perform complex calculations.
The strong efforts towards the realization of scalable quantum computing are motivated by the promise of super-polynomial speed-up in several computational tasks. A first fundamental step in this route is to provide concrete evidence of ‘quantum supremacy'. Such a regime is obtained when a quantum device is capable of solving a specific problem faster than the best classical hardware. A serious attempt to provide a first demonstration of quantum speed-up with affordable resources is represented by boson sampling [1,2], a non-universal quantum-computational protocol exploiting indistinguishable single photons and linear optical devices. In this context, strong theoretical evidence has been reported that the interference pattern generated by a boson sampler is hard to simulate with classical resources, this pattern being related to the calculation of a hard-to-compute quantity known as a matrix permanent. Formal achievement of the regime of quantum supremacy will answer two very relevant questions: Are we able to benefit from the power of quantum mechanics or does there exist a fundamental reason why this achievement is forbidden? Once this regime has been achieved, are we able to certify the correct functioning of the quantum boson samplers or do we face an undecidability problem? Starting from these intriguing and promising theoretical results, an experimental race kicked off with the aim of reaching the quantum supremacy regime with boson sampling by employing a photonic platform [3]. A large amount of effort has been devoted to pushing the limits of current photonic systems [1], in order to significantly increase the size (number of photons and optical modes) that can be handled with a quantum device. In parallel to this quest for improved quantum performance, it is fundamental to benchmark the limits of a classical simulation of boson sampling. More specifically, in the race for quantum supremacy, what is the best possible classical simulation that can be performed? Such an analysis would be necessary to define the computational capacity possessed by such a classical agent to perform the task (see Fig. 1). In the race for quantum supremacy, a user who employs the best classical hardware and software, and a second agent who exploits a quantum device compete to solve the boson sampling task faster. In a recent article, Wu et al. [4] have provided a fundamental step to define and push the limits of a classical simulation of boson sampling. The authors have employed Tianhe-2, one of the world's most powerful super-computers, to perform such a task. More specifically, by using such high-performance computing hardware and by employing the fastest algorithm to date, Wu et al. have shown that calculating the permanent of a |50×50| matrix can be performed in 100 min. Recent results [5] have shown that this computation corresponds to the classical hardness of drawing a 50-photon sample from a boson sampling distribution. The strong computational capacity of Tianhe-2 also allowed us to address the precision that the known fastest classical algorithms can reach in such a large-scale calculation. This latter aspect is indeed of fundamental importance, since any reliable classical computation must be accompanied by an appropriate error analysis. The study reported in Ref. [4] represents by far the largest calculation to date related to the classical simulation of boson sampling, whose relevance can be found in two different but related aspects. First, it raises the bar for any quantum implementation aimed at reaching quantum advantage, and provides a fundamental benchmark to which any photonic hardware must compare. This result suggests that a larger effort must be dedicated to pushing quantum technologies in this direction. Second, improved classical simulation also has important consequences in the context of boson sampling certification, i.e. the verification that a quantum device is correctly performing the required task. Indeed, first validation procedures in the quantum advantage regime must be assisted by powerful classical computational resources that, while requiring longer processing time than quantum hardware, will necessarily have to perform complex calculations.
In his famous 1982 lecture, Nobel laureate Richard Feynman seeded the idea of quantum computation. He reasoned that the amount of classical information required to describe a quantum system scaled exponentially with its size, and thus could not be efficiently simulated by a classical computer. Indeed, a fairly modest system of 100 quantum bits would already require more information to describe than all the present data storage on Earth [1]. Given this, there would likely be computational tasks that achieve quantum supremacy—impossible to perform practically without the aid of a quantum information processor. In the 37 intervening years, these initial ideas have solidified, as the discovery of a string of quantum algorithms—from efficient factoring of large numbers to simulating chemical reactions—fueled a race for quantum supremacy. Yet, even theoretically, conclusively deciding—if quantum supremacy has been achieved—is non-trivial. Judging what problems can be efficiently solved by classical computers turns out to be exceedingly hard in itself. Take factoring, for instance. It is really classical intractable, or is it merely that we have not yet found an efficient classical means to factor? As we better understood how quantum computers process information, we also got better at finding classical methods to simulate them. This in turn has inspired new efficient classical algorithms in surprising situations [2]. Prof. Man-Hong Yung, associate professor at the Southern University of Science and Technology, and colleagues provided a beautiful illustration of such a discovery. Their work has been published in the July 2019 issue of National Science Review [3]. It considers a specialized type of linear optical quantum processor. The input and output are encoded within an array of optical modes by adjusting their initial photon number. Computation is achieved by interfering with these modes via a network of beam splitters, such that photons in one mode are transferred to another. Such processors have captured scientific attention due to their ability to perform a task known as boson sampling. When each mode is initialized with a single photon, the output statistics allows efficient evaluation of the matrix permanent—a computational problem with very strong evidence for being classically intractable [4]. This, coupled with their relative ease of engineering, made this specialized type of quantum processor an ideal holy grail for demonstrating quantum supremacy. Given this, one may expect that classical computers can say little about what such processors will output. After all, the number of paths that a photon can take to reach one mode from another scales exponentially with the size of the beam-splitter network. One may therefore assume that it would take exponential resources to track these paths to determine the probability of a photon moving from mode A to mode B. However, Yung and colleagues showed that a classical algorithm can efficiently estimate the transition probabilities—as long as one allows for additive error. This result represents a significant generalization of prior results by Scott Aaronson, which considers the case where there exists at most one photon in each mode [5]. In contrast, Yung's solution works regardless of the number of photons encoded in each mode. The result not only benefits the simulation of bosonic interference, but also has strong consequences in computing. Suppose that we use such specialized quantum processors to make binary decisions. That is, depending on input, we feed in particular distributions of photons—and then decide to take a particular action depending on whether the probability of seeing x photons in a particular output is above some threshold. In such contexts, a classical algorithm that estimates this probability up to the bounded additive error is sufficient for making identical decisions as the quantum processer itself. As such, Yung's algorithm enables a classical computer to be able to efficiently mimic the decisions of its specialized linear optical counterpart. This will no doubt have important ramifications in the present race for quantum supremacy. Much present effort is spent on demonstrating quantum supremacy through boson sampling, and the results here indicate that we will need to introduce more sophisticated quantum architectures if we wish to demonstrate such supremacy in making the right decisions.
Boson sampling, thought to be intractable classically, can be solved by a quantum machine composed of merely generation, linear evolution and detection of single photons. Such an analog quantum computer for this specific problem provides a shortcut to boost the absolute computing power of quantum computers to beat classical ones. However, the capacity bound of classical computers for simulating boson sampling has not yet been identified. Here we simulate boson sampling on the Tianhe-2 supercomputer, which occupied the first place in the world ranking six times from 2013 to 2016. We computed the permanent of the largest matrix using up to 312 000 CPU cores of Tianhe-2, and inferred from the current most efficient permanent-computing algorithms that an upper bound on the performance of Tianhe-2 is one 50-photon sample per ∼100 min. In addition, we found a precision issue with one of two permanent-computing algorithms.
Quantum computation was envisioned by Feynman as a valuable means of solving quantum problems [1]. The question is, how do we prove the superiority of quantum computing over classical devices? A common misconception in describing the power of a quantum computer over a classical computer is that quantum bits can be prepared in a superposition of an exponential number of states, which cannot be achieved with classical bits. This statement is certainly true, but it does not tell us, specifically, what problems quantum computers can efficiently solve while classical computers fail to do so. In fact, based exactly on the technique of Feynman's path integral, the output (i.e. transition amplitude) of any quantum circuit can be calculated classically with a polynomial amount of memory (although the temporal cost would be exponential). Therefore any quantum computation can be simulated classically; the key question is about the efficiency. Much effort has been made trying to pin down the power of quantum computers based on the machinery of the computational complexity theory. In particular, the class of computational (decision) problems that can be solved efficiently by a quantum computer is called BQP (bounded-error quantum polynomial time); the counterpart for classical computers is called P (polynomial time). Of course, BQP cannot be less than P; in principle quantum computers can simulate classical computers efficiently. The point is that the foundation of quantum computation cannot be established without a proof showing that BQP is strictly larger than P (i.e. BQP ≠ P). In this sense, this question is as interesting as the famous challenge of proving P ≠ NP (non-deterministic polynomial time). A well known instance of an NP problem is the factoring problem; Shor's quantum algorithm is capable of solving the factoring problem in polynomial time, which is not achievable with the best classical algorithm discovered so far. However, we still cannot rigorously exclude the possibility of the existence of an efficient classical algorithm for factoring. Although again without a proof, it is commonly believed that quantum computers are unable to solve some of the NP problems. In fact, the proof of this statement (if true) provides a possible path to prove P ≠ NP. A potentially less demanding question would be, when do we expect a quantum computer to be able to perform some well defined tasks (not necessarily related to any practical problem) that cannot be simulated with any currently available classical device, within a reasonable time? Once achieved, the status is called ‘quantum supremacy' [2]. One may ask, what about Grover's search algorithm, which provides a quadratic speedup over the classical search? Can we say that we can already achieve quantum supremacy with Grover's algorithm? The problem is that, instead of taking the large-N limit, quantum supremacy requires us to determine the actual number of qubits and gates that can no longer be simulated by classical computers within a reasonable time. To elaborate further, we remark that there can be two notions of classical simulation, namely ‘strong simulation' versus ‘weak simulation'. Strong simulation refers to the cases of calculating the transition probabilities, or expectation values of observables, to a high accuracy in polynomial time. Weak simulation requires the probability distributions to be accurately reproduced, which involves ‘sampling' the different outcomes of the quantum devices. Some quantum circuits that cannot be strongly simulated with an efficient classical means may have efficient classical methods for weak simulation [3]. However, we should be careful about the accuracy requirement in the simulation tasks. For example, for many decision problems, it may be sufficient to determine the transition amplitudes to within an additive error, instead of multiplicative error. In other words, the classical simulability of quantum computational problems depends on the accuracy requirement. Currently, there are three popular approaches for achieving quantum supremacy (see Fig. 1), namely (i) boson sampling [4], (ii) sampling IQP (instantaneous quantum polynomial) circuits [5], and (iii) sampling chaotic quantum circuits [6]. In all of these approaches, distributions of different bit strings or photon numbers are sampled from the quantum devices. Furthermore, they all involve the assumption that classical computers are unable to efficiently determine the transition amplitudes, and/or reproduce (or approximate) the distributions of quantum devices performing these tasks, in the sense of both strong and weak simulations. Three different approaches for achieving quantum supremacy: (a) boson sampling; (b) IQP circuits; (c) chaotic quantum circuits. In boson sampling, single photons are injected into different modes of a linear optical network. The task is to determine the photon distributions at the output ports. The key feature of boson sampling is that the transition amplitudes are related to the permanents of complex matrices, which are in general very difficult to calculate exactly, or approximate to within an multiplicative error; these problems belong to the #P-hard complexity class. Moreover, efficient weak simulation of boson sampling is believed to be impossible, unless the polynomial hierarchy collapses to its third level. However, a recent numerical study suggests that, in order to achieve quantum supremacy with boson sampling, one needs to simultaneously generate at least 50 or more single photons, which is still highly technologically challenging. An interesting question is whether linear optics, in the setting of boson sampling, can be applied to solve decision problems. In this case, the transition amplitudes may need to be determined to an additive error. However, we found that a large class of decision problems associated with boson sampling can be simulated by a classical computer [7], settling an open problem [8] by Aaronson and Hance. IQP circuits represent a simplified quantum-circuit model. The initial state starts from the all-zero state, |000…0〉, followed by applying Hadamard gates to each qubit. Then, diagonal gates are applied to the qubits, followed by applying Hadamard gates to each qubit again. The argument of showing the complexity in simulating IQP circuits is similar to that of boson sampling. In fact, the complexity of boson sampling was inspired by the complexity results of IQP circuits. However, when subject to noise, IQP circuits may become classically simulable [9]. Lastly, chaotic quantum circuits refer to the class of quantum circuits where two-qubit gates are applied regularly, while single-qubit gates are applied randomly from a gate set. The output distributions of these circuits approach the so-called Porter–Thomas distribution, which is a characteristic feature of quantum chaos. Recently, there have been several numerical investigations aiming to explore the ultimate limit of classical computing in simulating low-depth quantum circuits. For the ideal cases, one needs to consider both the qubit number and circuit depth for benchmarking quantum supremacy. However, in the presence of noise, these chaotic circuits may also become classically simulable for high-depth circuits [10]. Finally, other than these three approaches, one should expect that quantum supremacy can be achieved for many practical applications, e.g. simulation of quantum chemistry or quantum machine learning. When will it happen? Only time will tell!
Boson sampling represents a promising approach to obtain evidence of the supremacy of quantum systems as a resource for the solution of computational problems. The classical hardness of Boson Sampling has been related to the so called Permanent-of-Gaussians Conjecture and has been extended to some generalizations such as Scattershot Boson Sampling, approximate and lossy sampling under some reasonable constraints. However, it is still unclear how demanding these techniques are for a quantum experimental sampler. Starting from a state of the art analysis and taking account of the foreseeable practical limitations, we evaluate and discuss the bound for quantum supremacy for different recently proposed approaches, accordingly to today's best known classical simulators.
Boson sampling is a computational problem that has recently been proposed as a candidate to obtain an unequivocal quantum computational advantage. The problem consists in sampling from the output distribution of indistinguishable bosons in a linear interferometer. There is strong evidence that such an experiment is hard to classically simulate, but it is naturally solved by dedicated photonic quantum hardware, comprising single photons, linear evolution, and photodetection. This prospect has stimulated much effort resulting in the experimental implementation of progressively larger devices. We review recent advances in photonic boson sampling, describing both the technological improvements achieved and the future challenges. We also discuss recent proposals and implementations of variants of the original problem, theoretical issues occurring when imperfections are considered, and advances in the development of suitable techniques for validation of boson sampling experiments. We conclude by discussing the future application of photonic boson sampling devices beyond the original theoretical scope.
Photonics is a promising platform for demonstrating quantum computational supremacy (QCS) by convincingly outperforming the most powerful classical supercomputers on a well-defined computational task. Despite this promise, existing photonics proposals and demonstrations face significant hurdles. Experimentally, current implementations of Gaussian boson sampling lack programmability or have prohibitive loss rates. Theoretically, there is a comparative lack of rigorous evidence for the classical hardness of GBS. In this work, we make significant progress in improving both the theoretical evidence and experimental prospects. On the theory side, we provide strong evidence for the hardness of Gaussian boson sampling, placing it on par with the strongest theoretical proposals for QCS. On the experimental side, we propose a new QCS architecture, high-dimensional Gaussian boson sampling, which is programmable and can be implemented with low loss rates using few optical components. We show that particular classical algorithms for simulating GBS are vastly outperformed by high-dimensional Gaussian boson sampling experiments at modest system sizes. This work thus opens the path to demonstrating QCS with programmable photonic processors.